Principal function.
In the investigations next to be described a more extended meaning is
given to the symbol [delta]. We will, in the first instance, denote by
it an infinitesimal variation of the most general kind, affecting not
merely the values of the co-ordinates at any instant, but also the
initial and final configurations and the times of passing through
them. If we put
_
/t'
S = | (T - V)dt, (1)
_/t
we have, then,
_
/t'
[delta]S = (T' - V')[delta]t' - (T - V)[delta]t + | ([delta]T - [delta]V)dt
_/t
_ _
| |t'
= (T' - V')[delta]t' - (T - V)[delta]t + |[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)| (2)
|_ _|t
Let us now denote by x' + [delta]x', y' + [delta]y', z' + [delta]z',
the final co-ordinates (i.e. at time t' + [delta]t') of a particle m.
In the terms in (2) which relate to the upper limit we must therefore
write [delta]x' - x`'[delta]t', [delta]y' - y`'[delta]t',
[delta]z' - z`'[delta]t' for [delta]x, [delta]y, [delta]z. With a
similar modification at the lower limit, we obtain
[delta]S = - H[delta][tau] + [Sigma]m(x`'[delta]x' + y`'[delta]y' + z`'[delta]z')
- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z), (3)
where H(= T + V) is the constant value of the energy in the free
motion of the system, and [tau](= t' - t) is the time of transit. In
generalized co-ordinates this takes the form
[delta]S = - H[delta][tau] + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (4)
Now if we select any two arbitrary configurations as initial and
final, it is evident that we can in general (by suitable initial
velocities or impulses) start the system so that it will of itself
pass from the first to the second in any prescribed time [tau]. On
this view of the matter, S will be a function of the initial and final
co-ordinates (q1, q2, ... and q'1, q'2, ...) and the time [tau], as
independent variables. And we obtain at once from (4)
dPS dPS \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (5)
dPS dPS |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /
dPS
and H = - -------. (6)
dP[tau]
S is called by Hamilton the _principal function_; if its general form
for any system can be found, the preceding equations suffice to
determine the motion resulting from any given conditions. If we
substitute the values of p1, p2, ... and H from (5) and (6) in the
expression for the kinetic energy in the form [Tau]' (see Sec. 1), the
equation